On Voiculescu's Semicircular Matrices

Abstract

Assume $\N$ is a von Neumann algebra of type II$_1$ with a tracial state $\tau_{\N}$, and $\M$ is the von Neumann algebra of the $n\times n$ matrices over $\N$ with the canonical tracial state $\tau_{\M}$. Let $\mathcal D_n$ be the subalgebra of $\M$ consisting of scalar diagonal matrices in $\M$. In this article, we study the properties of semicircular elements in $\M$ that are free from $\mathcal D_n$ with respect to $\tau_{\M}$. Then we define a new concept "matricial distance" of two elements in $\M$ and compute the matricial distance between two free semicircular elements in $\M$.